Abstract

[1] In this work, the transition at backward endfire between bound (nonleaky) and leaky modes on one-dimensional periodic structures printed on a grounded dielectric substrate is examined. This mode evolution has been characterized for a class of structures with a finite-width cross section. Such structures support leakage into surface waves as well as into space due to the presence of the substrate. The solutions in this transition region are calculated using an accurate full-wave moment-method approach that allows for the determination of both physical and nonphysical solutions, both of which play an important role in the transition region.

[3] One-dimensional periodic printed leaky-wave antennas (LWAs) are very attractive because they provide a simple means of obtaining a narrow fan beam at an angle θ0 (see Figure 1a) that scans with frequency from backward endfire (θ0 = −90°) to forward endfire (θ0 = 90°), passing through broadside (θ0 = 0°) [Oliner, 1963; Tamir, 1969]. In such an antenna the radiation is due to the periodicity, which produces an infinite number of space harmonics (Floquet waves), some of which may be fast while the rest are slow. Each fast space harmonic gives rise to a radiated beam, which is scanned angularly by varying the operating frequency. The radiation characteristics of any fast space harmonic can be conveniently represented in the form of dispersion curves on a Brillouin diagram, in which radiation generally occurs when the operating point is located within the fast-wave region [Oliner, 1963; Hessel, 1969; Baccarelli et al., 2007]. Of particular interest are regions in which transitions between modes of different type occur. Near transition regions, coupling may occur between different space harmonics of the same mode or between different modes. The first type of coupling is responsible for closed and open stopband regions, whereas the second type is responsible for the transitions observed at the end of the scanning regions, at backward and forward endfire. In this paper, we present a study concerning new and general features of the transition at backward endfire for a wide class of 1-D periodic printed LWAs.

[4] As is well known, 1-D open periodic structures can be divided in two main classes. The first class consists of structures surrounded exclusively by free space, for which radiation is allowed only into free space. One example of such a structure would be a periodic structure in free space, such as a helix. A second example would be an infinitely wide dielectric slab that has a permittivity that is periodically modulated along one direction. A third example would be a periodic set of infinitely wide metal strips printed on a grounded dielectric slab (Figure 1b). In the latter two cases, even though a dielectric layer is present, the fields have no variation along the direction perpendicular to the periodicity, and leakage may occur only into space, since this is the only medium that surrounds the structure. For all three of these examples, the transition regions at backward and/or forward endfire have been examined previously, depending on the case [Hessel and Oliner, 1963; Hessel, 1969; Majumder et al., 1997; Burghignoli et al., 2001]. However, it appears that no results have been presented for the transition region at backward endfire for the case of printed structures of infinite width (typified by the third example mentioned above).

[5] A second class consists of printed periodic structures of finite width, for which radiation is allowed into both the dielectric substrate (in the form of a surface wave) and into free space. An example would be a microstrip line with a periodic set of gaps along the length (Figure 1a). Radiation from uniform structures of finite width (e.g., a uniform microstrip line without gaps) has been investigated previously [Oliner and Lee, 1986a, 1986b; Oliner, 1987; Bagby et al., 1993; Nghiem et al., 1996; Mesa et al., 1999]. Less attention has been given to periodic printed structures of finite width [Baccarelli et al., 2006; Rodríguez-Berral et al., 2007], which is the subject of this investigation. At sufficiently high frequencies these structures can support leaky modes that not only can radiate into space, but also into surface waves that are guided by the dielectric substrate. There are frequency ranges in which the wave types can be bound (nonleaky), leaky into surface waves, or leaky into both surface waves and space. The transitions from one wave type into another can be complicated, and can even involve portions that are nonphysical. A new Brillouin dispersion diagram has been presented recently that describes graphically how these wave types are arranged as a function of frequency [Baccarelli et al., 2007]. This new Brillouin diagram is a generalization of the customary diagram [Oliner, 1963; Hessel, 1969]. Because the new diagram treats surface-wave leakage in addition to space leakage and bound modes, there are of course additional transition regions in the new Brillouin diagram. However, these new transition regions have so far not been investigated. In this paper, we focus our attention on the particular properties of the transition region between the fundamental bound mode and the lowest surface-leaky mode, which describes the transition at the end of the scanning region at backward endfire. Results are presented for a particular type of structure, but the conclusions should be general to a wide class of 1-D periodic LWAs printed on dielectric substrates.

[6] The class of structures considered here includes microstrip lines periodically modulated along the longitudinal direction (called z), but with a transverse finite width (i.e., with a finite size of the metallization in the y direction, see Figure 1a). 1-D periodic printed structures for which the radiation is allowed only into free space (such as the metal strip grating in Figure 1b) are also reviewed, so that the peculiar features of the transition region at backward endfire for finite-width structures may be better understood.

[7] The periodicity allows us to express the field quantities, by means of Floquet's theorem, as a sum of an infinite number of space harmonics with complex propagation wave numbers kzn = βn − jα = β0 + 2πn/p − jα, where p is the spatial period and n = 0, ±1, ±2, …. Each space harmonic has a different phase constant βn, but the same attenuation constant α. Furthermore, each space harmonic possesses different longitudinal phase velocities, while the group velocities of the individual space harmonics are identical and equal to the group velocity of the entire mode [Hessel, 1969]. A time-harmonic dependence ejωt is assumed and suppressed throughout.

[8] The metal strip grating on a grounded dielectric slab considered in this section is shown in Figure 1b with the relevant coordinate system; it consists of an infinite array of perfectly conducting metal strips of width w and negligible thickness, periodically arranged along the z direction with spatial period p, infinitely long in the y direction, placed on a grounded lossless dielectric slab of relative permittivity ɛr, relative permeability μr, and thickness h. Modal fields of the metal strip grating that propagate in the z direction (orthogonal to the strips) will be considered in the following. In this case, the two-dimensional nature of the problem allows us to decouple the TEx,z and TMx,z polarizations. Modes of the periodic structure may be regarded as the TE and TM modes of a grounded dielectric slab perturbed by the presence of the metal strip grating (when w < p/2) or, alternatively, as the TE and TM modes of a parallel plate guide perturbed by the presence of slots in the upper plate (when w > p/2).

[9] Structures that are infinite in the y direction, such as the metal strip grating of Figure 1b, allow for leakage (radiation) only into free space. The modal electric field may be expressed in the air region (i.e., x > 0) as

where

is the transverse complex wave number in the x direction for the nth space harmonic. The presence of a square root in (2) makes the transverse wave number a double-valued function of the modal propagation wave number kzn = βn − jα. Hence, a different branch choice in (2) is mathematically possible for each of the infinite number of space harmonics. That is, each space-harmonic wave number may be chosen to be proper (m{kxn} < 0, or αxn > 0, so that the space harmonic decays in the x direction) or improper (m{kxn} > 0, or αxn < 0, which corresponds to an exponential increase in the x direction). The proper or improper nature of each space harmonic in the infinite series in (1) determines the spectral character of the modal solution [Hessel, 1969]. Numerical results for this structure are provided, based on a full-wave moment-method discretization of the relevant electric field integral equation (EFIE) in the unit cell. The adopted numerical approach, originally developed by Baccarelli et al. [2002a] for the analysis of linear phased arrays of microstrip LWAs, has been adapted by Baccarelli et al. [2005b] to solve the homogeneous eigenvalue problem of determining surface waves and leaky waves propagating along an arbitrary oblique direction in the grating plane. A specific structure is considered, with parameters w = 1.8 mm, p = 4 mm, h = 1.27 mm, μr = 1, and ɛr = 6, as shown in Figure 1b. In Figure 2, the Brillouin diagrams of the fundamental (perturbed) TM0 mode, which propagates orthogonally to the strips, is shown in the region −2 < βp/π < 2; the n = 0 and n = −1 space harmonics are reported. The diagram is symmetric with respect to the vertical axes βp/π = 0, ±1. Black lines correspond to space harmonics with positive group velocity and gray lines are those with negative group velocity, which contribute to the modal field representation for positive and negative values of the z coordinate, respectively, for a structure excited by a source at z = 0. Solid lines correspond to space harmonics of the forward type, for which phase and group velocities are in the same direction, while dashed lines indicate space harmonics of the backward type, for which the phase and group velocities are in opposite directions [Hessel, 1969]. The TM0 mode has its cutoff at zero frequency and is bound at low frequencies, where the modal field is confined to the structure. In this low-frequency range, all the space harmonics are proper and lie inside the bound regions of the Brillouin diagram in Figure 2, i.e., within the triangles delimited by the gray thin dotted lines β = ±k0 and their periodic replicas. (The bound region denotes the region where a physical solution will be nonleaky.) The solution at low frequency corresponds to that of a slightly perturbed TM0 mode on a grounded slab. Within the bound regions the modal solution is proper real (i.e., proper with a real wave number), except at the closed stopband where the solution is proper complex (i.e., proper with a complex wave number) and the normalized phase constants of the space harmonics are characterized by βp/π = ±(2n + 1). Closed stopband regions are the consequence of contradirectional coupling between different space harmonics with group velocities of opposite signs [Hessel, 1969]. This is shown in Figure 2, where the dashed black and solid gray (or solid black and dashed gray) curves of the n = 0 and n = −1 space harmonics couple at the end of the first passband region (which extends over 0 < k0p/π < 0.53), thus giving rise to a closed stopband regime (for 0.53 < k0p/π < 0.66) where the normalized phase constant of the proper complex solutions lie exactly along the βp/π = ±1 vertical lines. Within the closed stopband region the attenuation constant corresponds to reactive decay (similar to what happens in a waveguide below cutoff) and not to leakage.

Figure 2.

Brillouin diagram for the n = 0 and n = −1 space harmonics of the fundamental TM0 mode of the structure shown in Figure 1b. The legend for the normalized phase constants is as follows: harmonics with positive group velocity (black lines); harmonics with negative group velocity (gray lines); forward waves (solid lines); backward waves (dashed lines). The β = ±k0 lines and their replicas are shown with thin gray dotted lines. The shadowed yellow area indicates the first Brillouin region.

[10] From the upper edge of the closed stopband a second passband region occurs where the modal solution is again proper and real (in the range 0.66 < k0p/π < 0.69). By increasing frequency, a bound mode is no longer possible. When the real proper solutions lie within the bound regions but near to one of its sides (β = ±k0), a transition to a backward leaky regime occurs [Hessel and Oliner, 1963; Hessel, 1969]. Here the solution becomes again proper complex, as a consequence of a coupling between two real solutions, with different group velocities, of the same space harmonic. This will be shown later in Figure 3.

[11] An alternative description of this transition region can be given by observing the behavior of the space harmonics only in the first Brillouin region, i.e., the shadowed region −1 < βp/π < 1. Let us consider only space harmonics with positive group velocity (black lines in Figure 2), if we are interested in the modal field representation for positive values of the z coordinate. At low frequencies, in the first passband regime, the space harmonic with positive group velocity is the n = 0 harmonic, which lies in the first bound-mode triangle (delimited by the dashed lines) in the positive half-space βp/π > 0. This space harmonic is represented by a black solid line in Figure 2 and represents a forward wave that is proper real. In the second passband, after the closed stopband, the n = −1 harmonic lying in the first bound-mode triangle in the negative half-space βp/π < 0 has a positive group velocity (it is represented by a black dashed line in Figure 2). This harmonic contributes to the field representation for positive values of the z coordinate in the second passband region and it is a backward wave that is proper real. By increasing frequency, this backward proper real solution gives rise to a complex proper solution, by merging with a forward proper real solution close to the β = −k0 edge of the triangle (this will be seen in Figure 3). This complex proper n = −1 harmonic becomes physical as the frequency increases by exiting the bound-mode triangle and entering the fast-wave region of the Brillouin diagram (i.e., the region between the β = ±k0 curves), where it represents a backward leaky wave. In particular, this backward leaky wave is responsible for radiation at an angle in free space in the backward direction when −k0 < β−1 < 0 [Oliner, 1963; Hessel, 1969; Tamir, 1969]. (The same description of this transition region can be provided if we follow space harmonics with negative group velocities, the gray lines in Figure 2, in the first Brillouin zone. In this case, the backward proper real n = −1 harmonic, the gray dashed line in Figure 2, gives rise to a complex proper solution near the β = +k0 edge of the first positive bound-mode triangle.)

[12] In Figure 3a a more detailed description of the transition region at backward endfire for the perturbed TM0 mode of the metal strip grating is provided concerning the behavior of the normalized phase constant of the n = −1 harmonic in the negative region of the Brillouin diagram. In Figure 3b the normalized attenuation constant versus normalized frequency is shown. In the second passband, above the frequency of the closed stopband regime (where the solution is proper complex with β−1p/π = −1, and is shown as a vertical black short-dashed line in Figure 3a), the n = −1 harmonic of two distinct proper real solutions (one a backward solution, denoted with a black dashed line, and the other a forward solution, denoted with a black solid line) coalesce at the splitting point (point S in Figure 3a), giving rise to a complex proper solution (black short-dashed line). By further increasing frequency, the proper complex solution exits the edge of the triangle, crossing the β = −k0 curve (point C in Figure 3a). When β−1 > −k0 the solution is in a physical radiative region (i.e., it directly contributes to the field representation) [Tamir and Oliner, 1963; Hessel, 1969]. Between points S and C, within the bound-mode triangle, the solution is nonphysical, since it lies within the bound-mode (slow-wave) triangle even though its attenuation constant is different from zero, as shown in Figure 3b.

[13] By decreasing frequency from the splitting point S, the forward proper real solution (the black solid line between points S and T in Figure 3a) becomes tangent to the β = −k0 curve and at the point T it changes its spectral nature, with the n = −1 space harmonic being improper for frequencies below this point. The solution thus evolves into a new improper real solution (black dashed-dotted line in Figure 3a), which lies in the bound-mode triangle, but is nonphysical. This new improper modal solution has been obtained by choosing the improper determination of the square root in (2) only for the n = −1 space harmonic (this is a nonphysical choice of the square root for this wave number).

[14] In Figure 3b the normalized attenuation constant of the perturbed TM0 mode is shown to be different from zero below and above the second passband region. High values of the normalized attenuation constant are observed within the closed stopband region at lower frequencies, while lower values of the normalized attenuation constant occur above the second passband, corresponding to frequencies above the splitting point frequency, where the mode is in the backward leakage region. We note that the modal behavior shown in Figure 3 is very different from the improper spectral gap that occurs between the leaky-wave and bound-wave ranges in the modal dispersion diagrams of many uniform open waveguides [see, e.g., Lampariello et al., 1990; Baccarelli et al., 2002b] and at the forward endfire of open periodic structures, such as the metal strip grating [see, e.g., Hessel and Oliner, 1963; Hessel, 1969; Majumder et al., 1997; Burghignoli et al., 2001]. The transition shown here in Figure 3a does not occur between improper real and improper complex solutions, but between proper real and proper complex solutions. The observed behavior in Figure 3 has some similarities with the transition observed in open gyrotropic and metamaterial structures that support backward surface and leaky waves [see, e.g., Baccarelli et al., 1997, 2005a] and can more properly be termed a proper spectral gap, since the modes involved in the transition are proper during the transition. This type of proper spectral-gap region is qualitatively the same as that observed for 1-D periodic structures in free space, such as a wire helix [Hessel, 1969] and periodically modulated slow-wave structures [Hessel and Oliner, 1963; Hessel, 1969]. It is likely to characterize the transition between a bound mode and a leaky mode at backward endfire for any 1-D periodic LWA that can radiate only into free space, whether or not a substrate is present.

[15] Modal propagation regimes on 1-D periodic printed structures of finite width, such as the periodically loaded microstrip line of Figure 1a, are analyzed here by using a rigorous formulation based on a spectral-domain approach and a mixed-potential form of the 1-D periodic Green's function [Baccarelli et al., 2006]. This type of structure allows for leakage (radiation) into free space, just as with the metal strip grating, and also for leakage into the substrate. The modal electric field in the air region has in this case the form of a continuous spectrum:

where kxn = (k02 − ky2 − kzn2)1/2 is the transverse wave number in the x direction for the nth space harmonic. For this type of structure surface-wave leakage is permitted in addition to space leakage because of the finite width of the metallization and the unbounded grounded substrate in the y direction. Different paths of integration can be chosen in (3) for each space harmonic, detouring around the branch points and surface-wave pole singularities in the ky spectral plane in a specified manner. The path of integration in (3) determines the nature of the leakage [Bagby et al., 1993; Mesa et al., 1999; Baccarelli et al., 2007]. This choice of path also allows for both physical and nonphysical solutions to be obtained, and this aspect is very important when examining the spectral-gap region, as this region is the transition between physical and nonphysical solutions (as demonstrated in Figure 3).

[16] The structure in Figure 1a can be seen either as a microstrip line on a grounded dielectric slab periodically perturbed by gaps or as a 1-D array of “dog-bone” patches on a grounded dielectric slab. A specific structure is considered here, with parameters p = 4 mm, h = 1.27 mm, μr = 1, and ɛr = 6, as shown in Figure 1a. The dispersion behavior of the n = 0 and n = −1 space harmonics of the fundamental mode supported by this structure, which is a perturbation of the quasi-TEM mode of the uniform microstrip line, is shown in the generalized Brillouin diagram [Baccarelli et al., 2007] in Figure 4. The same line styles used in Figure 2 to denote forward and backward space harmonics with positive and negative group velocities is adopted here. Starting at low frequencies, the mode is bound; the modal field is confined to the structure with all the space harmonics proper. This means that the spectral ky integration in (3) is carried out along the entire real axis on the top (proper) sheet of the complex ky plane for all the space-harmonic terms in the summation of (3), as shown in Figure 5a [Baccarelli et al., 2006]. The wave numbers of all space harmonics lie inside the bound (slow-wave) region of the generalized Brillouin diagram in Figure 4, where the mode is slow with respect to free space and also with respect to the TM0 mode of the grounded substrate. This is the shaded region within the triangles (which denote slow with respect to free space) that is also below the curves of the TM0 mode (β = ±k) of the substrate and their periodic replicas (thin gray solid lines) [Baccarelli et al., 2007]. Within the bound region the modal solution is proper real, except within the closed stopband, which extends over 0.392 < k0p/π < 0.588 (corresponding to βp/π = ±1 in Figure 4), where it is complex proper. Closed stopband regions are, as stated above for the metal strip grating case, the consequence of coupling between different space harmonics with opposite group velocities. Although the solution is complex in this region, the attenuation constant does not correspond to leakage, but to reactive decay.

Figure 4.

Generalized Brillouin diagram for the n = 0 and n = −1 space harmonics of the fundamental quasi-TEM mode of the structure shown in Figure 1a. The purely bound solutions now lie inside of the gray shaded regions, which are bounded by the β = ±k curves rather than the β = ±k0 lines as in Figure 2. We see that when a solution first emerges from the “bound region,” it enters a region in which radiation can occur in surface-wave form but not into space. The legend for the normalized phase constants of the harmonics is as follows: positive group velocity (black lines); negative group velocity (gray lines); forward waves (solid lines); backward waves (dashed lines). The β = ±k0 lines and their periodic replicas are shown with thin gray dotted lines. The β = ±k lines and their periodic replicas are shown with thin gray solid lines.

Figure 5.

Paths of integration in the complex ky plane for the n = −1 space harmonic in a spectral-domain analysis. The different paths correspond to different propagation regimes. The legend is as follows: Paths are shown with a gray solid line, TM0 pole singularities are represented by a cross, k0 branch points are represented by a dot, and the branch cuts are denoted by a wavy line. (a) Passband, closed stopband, or backward space- and/or surface-leaky regimes. Two situations are shown: in the passband the poles and branch points are on the imaginary axis, and in the other cases they are in the second and fourth quadrants. (b) Real improper solution. The poles and branch points are on the imaginary axis, and residue contributions from the poles are included in the calculation.

[17] By increasing frequency, a bound mode is no longer possible. Above the closed stopband, a second narrow passband arises and the wave numbers of all space harmonics are again proper real. This is the portion of the curves within the bound-mode (shaded) region that lie above the closed stopband region (i.e., the region for which βp/π ≠ ±1 and 0.588 < k0p/π < 0.595 in Figure 4). When the solution lies within the bound-mode region but near to one of its borders (β = ±k and their periodic replicas), a transition to a complex proper solution occurs. To better understand the nature of this transition, we focus on the behavior of the wave numbers of the space harmonics in the first Brillouin region, i.e., the region −1 < βp/π < 1. We also choose to follow the behavior of the space harmonics with a positive group velocity, since we are interested in the modal field contribution for positive values of the z coordinate. Within the second passband, the space harmonic with positive group velocity is the backward proper real n = −1 harmonic that lies in the bound region on the negative side of the diagram (the black dashed line that lies in −1 < βp/π < 0 in Figure 4). By increasing frequency, this backward proper real solution gives rise to a complex proper solution, by merging with a forward proper real solution, close to the β = −k edge of the bound region (this will be seen more clearly in Figure 6). This proper complex n = −1 harmonic is a backward surface-leaky wave that becomes physical, after crossing the β = −k curve and entering the backward surface-leakage region of the modified Brillouin diagram. This is the region between the β = −k curve (thin gray line) and the β = −k0 line (gray dotted line) in Figure 4. (The same description of this transition region can be provided if we follow space harmonics with negative group velocities in the first positive Brillouin zone. In this case, the backward proper real n = −1 harmonic (the gray dashed line lying within the positive bound-mode region 0 < βp/π < 1 in Figure 4) gives rise to a complex proper solution near the β = +k edge of the first positive bound-mode region.)

[18] In Figure 6 a more detailed description is provided of the transition region at backward endfire for the periodically loaded microstrip line on a grounded dielectric slab. The n = −1 harmonic of two distinct proper real solutions (a backward wave, denoted with a black dashed line, and a forward wave, denoted with a black solid line) coalesce at the splitting point S (shown in Figure 6a and also in a further enlarged scale in Figure 6c), giving rise to a complex proper solution (black short-dashed line). By further increasing frequency, the complex proper solution exits the edge of the bound region, crossing the β = −k curve (point C in Figures 6a and 6c). When the complex solution lies between the β = −k curve and the β = −k0 line, i.e., −k < β−1 < −k0, the solution is physical. Radiation then occurs into the TM0 surface wave of the grounded dielectric slab at an angle from the strip in the backward direction (backward surface-leaky regime).

[19] At higher frequency, the proper complex solution crosses the free-space line (thin gray dotted line for which β = −k0), thus leaving the backward surface-leaky region and now entering a region where the mode radiates into free space in the backward direction, as well as leaking into the surface wave in the backward direction (backward space + surface leaky regime). For the above-described regimes the correct integration path in the ky plane for all the space harmonics remains along the real axis on the top (proper) sheet, as shown in Figure 5a, consistent with the proper character of the modal solution [Baccarelli et al., 2007]. In Figure 6b the normalized attenuation constant of the perturbed quasi-TEM mode is seen to be different from zero below and above the second passband region. Higher values of the attenuation constant are observable below the second passband (within the closed stopband region) at lower frequencies, while lower values occur above the second passband (above the splitting-point frequency). When −k < β−1 the solution is a physical radiating mode. Between points C and D the mode leaks into the TM0 mode of the substrate, and beyond D the mode also radiates into free space (in the backward direction). Between the points S and C, within the bound region, the solution is nonphysical, since its attenuation constant is different from zero even though it is a slow wave with respect to both the TM0 surface wave and free space. By decreasing frequency from the splitting point S, one of the proper real solutions (i.e., the forward solution shown as a black solid line between points S and T in Figures 6a and 6c) becomes tangent to the β = −k curve and at the point T it changes its spectral nature, with the n = −1 space harmonic becoming improper. The solution thus evolves into a new improper real solution (the black dashed-dotted line in Figures 6a and 6c), which lies in the bound region but is nonphysical (since an improper real solution is always regarded as being nonphysical). This new improper modal solution has been obtained by using the integration path in the ky plane shown in Figure 5b only for the n = −1 space harmonic, and the real axis path on the top (proper) sheet (Figure 5a) for all other harmonics.

[20] We note that the modal behavior shown in Figure 6 is very different from the improper spectral gap which is found to occur between the surface-leakage and bound regimes in the modal dispersion diagrams of uniform open printed lines on a grounded dielectric slab [see, e.g., Shigesawa et al., 1993; Nghiem et al., 1996; Mesa et al., 2002], where improper modal solutions are involved in the transition. The reported behavior has instead some similarities with the transition analyzed in section 2 for 1-D open periodic structures, such as the metal strip grating in Figure 1b. However, the presence of a finite-width metallization on a grounded dielectric slab makes the overall modal spectrum of the periodically perturbed microstrip line in Figure 1a more intricate, due to the possibility of having a surface-leakage regime in addition to a space-leakage regime. In particular, the merging of solutions still takes place near the boundary of the bound-mode region, but this boundary is now where surface-wave leakage begins, rather than space-wave radiation. Furthermore, by increasing frequency, the character of the mode changes from leaking into only the surface wave to leaking into both the surface wave and into space, without the occurrence of any spectral-gap region. That is, the proper complex solution that is already complex due to leakage into the TM0 surface wave crosses the β = −k0 dotted line at point D in a smooth fashion, without merging with any other solution.

[21] The transition region described above corresponds to the end of the scanning region at backward endfire, and is qualitatively similar for a wide class of 1-D periodic LWAs printed on dielectric substrates, and for this reason it is particularly significant. It is well known that a fast space harmonic (e.g., the n = −1 harmonic) of a 1-D periodic LWA gives rise to a radiated beam at an angle θ0 in free space, which is defined by sin θ0 = β−1/k0 (see Figure 1a) and scans with frequency [Oliner, 1963; Tamir, 1969]. Values of the normalized phase constant in the range −1 < β−1/k0 < 0 correspond to radiation in the backward quadrant, i.e., from broadside (β−1/k0 = 0) to backward endfire (β−1/k0 = −1), by decreasing frequency. It is thus useful to provide a description of the same transition described in Figures 4 and 6 in terms of the phase and attenuation constants, both normalized with respect to the free-space wave number k0, as functions of frequency. This is provided in Figures 7a and 7b, respectively. Starting at 27 GHz and decreasing the frequency, the normalized phase constant of the n = −1 harmonic decreases until reaching β−1/k0 = −1 at 24.4 GHz (point D in Figure 7a) and correspondingly the radiated beam scans toward backward endfire. As the frequency is lowered from point D there is a region −k/k0 < β−1/k0 < −1, where leakage occurs in the dielectric substrate at an angle from the strip in the form of a backward surface-leaky wave, and the solution is still proper complex (see the normalized attenuation constant in Figure 7b). By further decreasing the frequency, the normalized phase constant, β−1/k0, crosses the −k/k0 curve and then splits into two proper real solutions, a backward solution and a forward solution, shown in Figures 7a and 7c as black dashed and black solid lines, respectively. For these solutions the leakage constant is zero in Figure 7b. As the frequency is lowered the forward proper real solution then becomes tangent to the −k/k0 curve and finally changes its spectral nature, becoming improper (a dashed-dotted line), whereas the backward proper real solution gives rise to a backward bound-mode solution down to the closed stopband frequency (the closed stopband is not seen on these expanded plots).

[22] We finally note that 1-D periodic LWAs optimized for radiating at backward endfire present narrower transition regions with comparison to the one shown here, requiring that the k curve be very close to the k0 line. The 1-D periodic microstrip structure in Figure 1a was chosen here because it has a relatively wide proper spectral gap, for the sake of clarity. However, the qualitative behavior of the spectral gap near backward endfire should be the same for all 1-D periodic printed LWAs.

4. Conclusions

[23] A thorough investigation has been presented here of the transition region at backward endfire for two major classes of 1-D periodic printed leaky-wave antennas. In one class the metallization of the printed antenna does not vary in the transverse direction, whereas in the other class the metallization has finite width. Examples of these two classes are shown in Figure 1 of this paper. In particular, a metal strip grating in which the strips are infinite in the transverse direction is shown in Figure 1b, whereas a periodically loaded microstrip line is presented in Figure 1a as an example of a structure with a finite-width metallization. There are some similarities between these two classes of structures, but there is a major difference. For those structures in which the metallization does not vary in the transverse direction, leakage can only occur via radiation into free space. For structures with a finite-width metallization radiation can still occur into free space but leakage may also occur into surface waves on the grounded dielectric substrate. The overall modal spectrum of the latter class of structures is therefore more complicated but also more interesting.

[24] In this paper, we present detailed analyses of the transition region at backward endfire for both of the structures shown in Figure 1, so that careful comparisons can be made between them. The structure for which the metallization does not vary in the transverse direction (Figure 1b) is examined carefully in section 2 of this paper. The example of the class for which the metallization is finite (Figure 1a), is investigated and discussed in detail in section 3 of this paper. A generalized Brillouin diagram is employed to explain why, as frequency is increased, the leakage occurs first in surface-wave form and then as a combination of surface-wave leakage and radiation into free space, and how physical and nonphysical solutions combine to evolve from one leakage form to another. The analyses in sections 2 and 3 are also compared with each other to indicate their similarities and differences.

[25] In conclusion, two major points should be stressed. The first point is that for the class of structures shown in Figure 1b (infinitely wide printed periodic structures), the transition region at backward endfire was observed to be qualitatively the same as that which occurs on periodic structures in free space (e.g., a wire helix), where there is no substrate, as well as other infinitely wide structures such as periodically modulated slabs. The second point is that for the class of structures shown in Figure 1a (structures of finite width), the transition region at backward endfire is of a different type, since the leakage mechanism is more complex. Although specific examples were chosen for the examination of each class of structure, the conclusions for each type of structure were discussed in general terms and apply to a wide class of 1-D periodic printed leaky-wave antenna structures.